{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:20:41Z","timestamp":1760242841793,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2016,12,16]],"date-time":"2016-12-16T00:00:00Z","timestamp":1481846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Ministry of Economy and Competitiveness","award":["TEC2016-78434-C3-3-R"],"award-info":[{"award-number":["TEC2016-78434-C3-3-R"]}]},{"name":"Spanish Ministry of Education","award":["IJCI-2014-19150"],"award-info":[{"award-number":["IJCI-2014-19150"]}]},{"name":"European Union 7th Framework Programme","award":["316861"],"award-info":[{"award-number":["316861"]}]},{"DOI":"10.13039\/100012818","name":"Comunidad de Madrid","doi-asserted-by":"publisher","award":["S2013\/ICE-2845"],"award-info":[{"award-number":["S2013\/ICE-2845"]}],"id":[{"id":"10.13039\/100012818","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the compression of a continuous real-valued source X using scalar quantizers and average squared error distortion D. Using lossless compression of the quantizer\u2019s output, Gish and Pierce showed that uniform quantizing yields the smallest output entropy in the limit     D \u2192 0    , resulting in a rate penalty of 0.255 bits\/sample above the Shannon Lower Bound (SLB). We present a scalar quantization scheme named lossy-bit entropy-constrained scalar quantization (Lb-ECSQ) that is able to reduce the     D \u2192 0     gap to SLB to 0.251 bits\/sample by combining both lossless and binary lossy compression of the quantizer\u2019s output. We also study the low-resolution regime and show that Lb-ECSQ significantly outperforms ECSQ in the case of 1-bit quantization.<\/jats:p>","DOI":"10.3390\/e18120449","type":"journal-article","created":{"date-parts":[[2016,12,16]],"date-time":"2016-12-16T10:55:28Z","timestamp":1481885728000},"page":"449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Entropy-Constrained Scalar Quantization with a Lossy-Compressed Bit"],"prefix":"10.3390","volume":"18","author":[{"given":"Melanie","family":"Pradier","sequence":"first","affiliation":[{"name":"Universidad Carlos III de Madrid, Madrid 28911, Spain"},{"name":"Gregorio Mara\u00f1\u00f3n Health Research Institute, Madrid 28007, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4276-5178","authenticated-orcid":false,"given":"Pablo","family":"Olmos","sequence":"additional","affiliation":[{"name":"Universidad Carlos III de Madrid, Madrid 28911, Spain"},{"name":"Gregorio Mara\u00f1\u00f3n Health Research Institute, Madrid 28007, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fernando","family":"Perez-Cruz","sequence":"additional","affiliation":[{"name":"Universidad Carlos III de Madrid, Madrid 28911, Spain"},{"name":"Stevens Institute of Technology, Hoboken, NJ 07030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1109\/TIT.1968.1054193","article-title":"Asymptotically efficient quantizing","volume":"14","author":"Gish","year":"1968","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1109\/TIT.1967.1054003","article-title":"Analog source digitization: A comparison of theory and practice (Corresp.)","volume":"13","author":"Goblick","year":"1967","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1109\/TIT.1984.1056920","article-title":"Optimum quantizer performance for a class of non-Gaussian memoryless sources","volume":"30","author":"Farvardin","year":"1984","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1109\/18.532878","article-title":"Efficient scalar quantization of exponential and Laplacian random variables","volume":"42","author":"Sullivan","year":"1996","journal-title":"IEEE Trans. Inf. 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